19.3 Anti-derivatives by Inspection and Classes of Functions that we can Anti-differentiate

The process of
differentiating
a known standard function can be achieved by parsing the function definition, and applying the appropriate rules of differentiation to each step in its definition.

There are no specific
anti-differentiation
rules; we have to use the differentiation rules backwards. And these allow us to anti-differentiate large classes of standard functions, but not all of them.

We can recognize some functions almost immediately as derivatives of others. For example, we can anti-differentiate any polynomial in
x
by applying the rule to differentiate powers backwards.

Thus an anti-derivative of
5x2
is
53x3
. (When anti-differentiating polynomials, a standard error is to suffer a mental lapse in the middle of doing this and differentiate instead of anti-differentiating some term; always check that you haven't done this.)

We can similarly anti-differentiate
any power
(even
1x
, one of whose anti-derivatives is
ln⁡(x)
). And we can anti-differentiate
eax
and
any polynomial in
sin⁡(x) andcos⁡(x)
, although doing the latter requires applying some tricks.

It turns out, that there are more tricks which allow us to differentiate
any
rational function
p(x)q(x) where
p and
q are polynomials, if we can factor
q(x) into linear and quadratic factors;
and tricks which allow us anti-differentiate
an exponential
times a polynomial, inverse powers of sines and cosines, functions that are
polynomials divided by the square root of a linear or quadratic functions, products
of exponentials sines and polynomials,
and various other classes of functions.

On the other hand there are simple looking standard functions that we cannot anti-differentiate, because their anti-derivatives do not happen to be standard functions.

(If these anti-derivatives were sufficiently important we would give them names, make tables of them and add them to the list of our basic functions (which consists now of
x,sin⁡x
and
ex
). As it happens they are important but not that important; we give them names and tabulate them, but call them
special
functions.

The simplest functions that cannot be anti-differentiated as standard functions are
exx
and
e−x2
both of which have special functions as specific anti-derivatives.)

There used to be a difference of opinion among teachers of calculus as to whether students should be made to learn all the tricks of anti-differentiation so that they could anti-differentiate anything that could be anti-differentiated.

Some felt that the mark of mastery of calculus was to be able to anti-differentiate strange looking functions by attempting to apply the various tricks. Others pointed out that there are "integral tables" which list essentially all anti-differentiable functions and their anti-derivatives, so that this skill, whose development is a wonderful exercise of ingenuity and memory, has very limited practical value.

At that time there were obstacles that students faced in getting a copy of a reasonably complete integral table, and in looking things up in huge ones, that gave the edge by a small margin to the first viewpoint.

Today, however, there are commercially available programs, Maple, MATLAB, and Mathematica (and perhaps others), which give you detailed formulae for the anti-derivative of any function you care to enter, and do so instantly; which seems to tilt the argument toward the second viewpoint.

Be that as it may, every student of calculus should be aware of the four basic tools of anti-differentiation, which we now discuss.

Before doing so we should notice the important fact that the action of finding an anti-derivative is a linear operation (as differentiating is) so that
an
anti-derivative of a sum is the sum of anti-derivatives of its summands.